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Theorem bcthlem2 25074
Description: Lemma for bcth 25078. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)
Hypotheses
Ref Expression
bcth.2 𝐽 = (MetOpenβ€˜π·)
bcthlem.4 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
bcthlem.5 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
bcthlem.6 (πœ‘ β†’ 𝑀:β„•βŸΆ(Clsdβ€˜π½))
bcthlem.7 (πœ‘ β†’ 𝑅 ∈ ℝ+)
bcthlem.8 (πœ‘ β†’ 𝐢 ∈ 𝑋)
bcthlem.9 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
bcthlem.10 (πœ‘ β†’ (π‘”β€˜1) = ⟨𝐢, π‘…βŸ©)
bcthlem.11 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
Assertion
Ref Expression
bcthlem2 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Distinct variable groups:   π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝐢,π‘Ÿ,π‘₯   𝑔,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧,𝐷   𝑔,𝐹,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝐽,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝑀,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   πœ‘,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   π‘₯,𝑅   𝑔,𝑋,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧
Allowed substitution hints:   πœ‘(𝑔)   𝐢(𝑧,𝑔,π‘˜,𝑛)   𝑅(𝑧,𝑔,π‘˜,𝑛,π‘Ÿ)
Proof of Theorem bcthlem2
StepHypRef Expression
1 bcthlem.11 . . . . 5 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
2 fvoveq1 7435 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘”β€˜(π‘˜ + 1)) = (π‘”β€˜(𝑛 + 1)))
3 id 22 . . . . . . . 8 (π‘˜ = 𝑛 β†’ π‘˜ = 𝑛)
4 fveq2 6892 . . . . . . . 8 (π‘˜ = 𝑛 β†’ (π‘”β€˜π‘˜) = (π‘”β€˜π‘›))
53, 4oveq12d 7430 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘˜πΉ(π‘”β€˜π‘˜)) = (𝑛𝐹(π‘”β€˜π‘›)))
62, 5eleq12d 2826 . . . . . 6 (π‘˜ = 𝑛 β†’ ((π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ↔ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›))))
76rspccva 3612 . . . . 5 ((βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
81, 7sylan 579 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
9 bcthlem.9 . . . . . 6 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
109ffvelcdmda 7087 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))
11 bcth.2 . . . . . . 7 𝐽 = (MetOpenβ€˜π·)
12 bcthlem.4 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
13 bcthlem.5 . . . . . . 7 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
1411, 12, 13bcthlem1 25073 . . . . . 6 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
1514expr 456 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))))
1610, 15mpd 15 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
178, 16mpbid 231 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))
18 cmetmet 25035 . . . . . . . . . . . 12 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
1912, 18syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐷 ∈ (Metβ€˜π‘‹))
20 metxmet 24061 . . . . . . . . . . 11 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2119, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2211mopntop 24167 . . . . . . . . . 10 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2321, 22syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐽 ∈ Top)
24 xp1st 8010 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋)
25 xp2nd 8011 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ+)
2625rpxrd 13022 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)
2724, 26jca 511 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*))
28 blssm 24145 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
29283expb 1119 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
3021, 27, 29syl2an 595 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
31 df-ov 7415 . . . . . . . . . . . 12 ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
32 1st2nd2 8017 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (π‘”β€˜(𝑛 + 1)) = ⟨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
3332fveq2d 6896 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩))
3431, 33eqtr4id 2790 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3534adantl 481 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3611mopnuni 24168 . . . . . . . . . . . 12 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3721, 36syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
3837adantr 480 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ 𝑋 = βˆͺ 𝐽)
3930, 35, 383sstr3d 4029 . . . . . . . . 9 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽)
40 eqid 2731 . . . . . . . . . 10 βˆͺ 𝐽 = βˆͺ 𝐽
4140sscls 22781 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
4223, 39, 41syl2an2r 682 . . . . . . . 8 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
43 difss2 4134 . . . . . . . 8 (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
44 sstr2 3990 . . . . . . . 8 (((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4542, 43, 44syl2im 40 . . . . . . 7 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4645a1d 25 . . . . . 6 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))))
4746ex 412 . . . . 5 (πœ‘ β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))))
48473impd 1347 . . . 4 (πœ‘ β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4948adantr 480 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
5017, 49mpd 15 . 2 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
5150ralrimiva 3145 1 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βˆ– cdif 3946   βŠ† wss 3949  βŸ¨cop 4635  βˆͺ cuni 4909   class class class wbr 5149  {copab 5211   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7976  2nd c2nd 7977  1c1 11114   + caddc 11116  β„*cxr 11252   < clt 11253   / cdiv 11876  β„•cn 12217  β„+crp 12979  βˆžMetcxmet 21130  Metcmet 21131  ballcbl 21132  MetOpencmopn 21135  Topctop 22616  Clsdccld 22741  clsccl 22743  CMetccmet 25003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9440  df-inf 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-n0 12478  df-z 12564  df-uz 12828  df-q 12938  df-rp 12980  df-xneg 13097  df-xadd 13098  df-xmul 13099  df-topgen 17394  df-psmet 21137  df-xmet 21138  df-met 21139  df-bl 21140  df-mopn 21141  df-top 22617  df-topon 22634  df-bases 22670  df-cld 22744  df-cls 22746  df-cmet 25006
This theorem is referenced by:  bcthlem3  25075  bcthlem4  25076
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